notes7b

notes7b - Notes 7 Parameter estimation : In general, for...

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Unformatted text preview: Notes 7 Parameter estimation : In general, for AR(p), MA(q) and ARMA(p,q) models, what we want to estimate are the series mean μ , the AR parameters φ ’s, the MA parameters θ ’s and the error variance σ 2 a . Estimation methods: 1. Method of moment 2. Method based on Least squares estimation 3. Method based on Unconditional least squares estimation 4. Maximum likelihood estimation 1. Method of moment The method consists of equating sample moments to theoretical moments and solving the resultant equations for obtaining estimates of the unknown parameters. (a) For time series mean μ , the estimator is the sample mean ¯ Z . (b) For AR(p), from Yule Walker equations         ρ 1 ρ 2 . . ρ p         =         ρ ρ 1 . . ρ p- 1 ρ 1 ρ ρ 1 . ρ p- 2 . . . . . . . . . . ρ p- 1 ρ p- 1 . . ρ                 φ 1 φ 2 . . φ p         . Replace ρ k ’s by r k ’s and then solve for ˆ φ 1 , ˆ φ 2 , . . . , ˆ φ p in terms of r 1 , r 2 , . . . , r p . Example. For an AR(1) process. Since ρ 1 = φ 1 , we have ˆ φ 1 = r 1 . Example. For an AR(2) process. Since ρ 1 = φ 1 + ρ 1 φ 2 and ρ 2 = ρ 1 φ 1 + φ 2 . 1 Therefore ˆ φ 1 = r 1 (1- r 2 ) 1- r 2 1 and ˆ φ 2 = r 2- r 2 1 1- r 2 1 . (c) MA(q) - the method of moments may cause problem. For example, let us consider a MA(1) process, recall that ρ 1 =- θ 1 + θ 2 ρ 1 θ 2 + θ + ρ 1 = 0 θ =- 1 ± q 1- 4 ρ 2 1 2 ρ 1 . Therefore ˆ θ =- 1 ± q 1- 4 r 2 1 2 r 1 ....
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.

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notes7b - Notes 7 Parameter estimation : In general, for...

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